Many schemes [1]}, [2]} estimate the temporal basis functions \(\mathbf {V}\)  from k-space navigators, followed by the estimation of the spatial factors \(\mathbf {U}\)  in (REF ). By contrast, the joint estimation of \(\mathbf {U}\)  and \(\mathbf {V}\)  from the measured data offers improved performance and reduces the need for specialized acquisition schemes with k-space navigators. The joint approaches pose the recovery of the signals from the undersampled measurements \(\mathcal {A}(\mathbf {X})\)  as:
\(\lbrace \mathbf {U}^*,\mathbf {V}^*\rbrace  = \arg \min _{\mathbf {U},\mathbf {V}} \left\Vert \mathcal {A}(\mathbf {U}\mathbf {V}^T) -\mathbf {B}\right\Vert ^2 + \lambda _1 \mathcal {R}_1(\mathbf {U}) + \lambda _2 \mathcal {R}_2 (\mathbf {V})\) 
